Differential vs derivative explained

Really there is no problem to go with this question just something that I was discussing awhile back but was never really cleared up, what is the difference between:

[tex] d(\int^{x}_{a} f(t)dt) = f(x) dx [/tex]

and

[tex] \frac{d}{dx}\int^{x}_{a} f(t) dt = f(x) [/tex]

I understand that the top equation is called the differential form and the bottom is the derivative form and really thats about it. I was told that they were different by a factor of dx such that:

[tex] df = \frac{d}{dx}*dx [/tex]

but I really don't understand how that makes any difference because if dx is infinitely small how does that affect the answer of the problem?

This arose from the book that I am using introducing The Fundamental Theorem of Calculus with the differential form and not the derivative form which evidently is the norm. I really just want a little clarification on this issue and apologize in advance if this is something quite obvious.

ok well your reply implies that it somehow effects my answers even on basic equations in calculus to use differential definitions over derivative, is that what you meant? If not then why should I get a different book if comes out the same?

Just use the definition:
[tex]
\frac{df}{dx}\Bigg|_{x=a}=f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}
[/tex]
this is what you seem to be calling the derivative form. Everything follows from the equation above. Don't worry about differentials.

ok, that is the very definition that my book illustrates but without the use of limits which are subbed for hyperreal numbers. Also I called that form the derivative form because either Dick or Mark44 called it that, is that not right?